Meaning of Constant Speed and Implications for Graphs

Logic of the Lesson

Shannon Coombs and Pat Thompson

 

The following is a lesson logic for teaching how to reason about situations involving initial conditions, constant rate of change, and different ways to represent those situations.

 

A lesson logic is the outline of how you will develop the lesson's main ideas. It does not pay attention to time, meaning that the "lesson" may transcend several class periods. It does not give the level of detail that a lesson plan gives, meaning it might not say how you will organize the classroom, how you will transition from one activity to another, etc. Instead, it focuses on the ideas you will develop, the way you develop them, and why you take the approach you take.

 

The following lesson logic provides a structure in which the surrounding conversation unfolds these ideas:

 

1.     Constant rate of change of quantity A with respect to quantity B means that every change in A is the same multiple of the corresponding change in B.

2.     One can represent simultaneous changes in two variables with a "calculus triangle" .

3.     When two quantities change at a constant rate with respect to each other, all calculus triangles representing changes in those quantities are similar.

4.     Ideas 1-3 together allow one to deduce that

a.     the graph of two variables changing at a constant rate is a line

b.     a (non-vertical) line in the X-Y coordinate system is the graph of two variables that change at a constant rate.

Meanings or schemes students must have before the lesson:

 

1)    Covariation of one quantity with another

2)    Finger tool thinking (covariation within a coordinate system; graphs are made of particles of pixie dust)

3)    Coordinates of a pixie dust particle tell the states of two variables simultaneously.

Steps in the Lesson Logic

 

Step

Action

Reason

1)     

What does it mean that someone travels at a constant speed? (Anticipate "speed never changes")

-      True. What does that mean about distance and time?

-      (Anticipating difficulty saying it) Suppose we are told how far this person traveled in some 16 second time period. What do we now know?

-   This 16 second period or other 16 second periods, too?

-   This tells us about how far he will travel in 8 seconds, 4 seconds, 1 second, 1 minute, 1 hour (and reason them out)

-      End with this summary: To say that a person travels at Z mi/hr means that however many hours he travels, he will travel Z times as many miles as that number of hours.

 

2)     

Set up scenario of traveling from home to store. Ask how to represent graphically that I am 3.8 miles from home and my watch says it is 7 minutes since I first looked at it.

 

3)     

I am traveling at the constant speed of ½ mile per minute (what does that mean?). I am at 3.8 miles and I first looked at my watch 7 minutes ago. How much farther will I travel in the next 1/10 minute? How much farther will I travel in the next 3/10 minute? How much farther will I travel in the next 10 minutes? (Remind them of the constant multiplier meaning of constant rate.)

 

4)     

Let's show these changes graphically (do the calculus triangle thing). Be sure to talk about changes in number of minutes and changes in number of miles. Convey that all values of time have a corresponding number of miles

 

5)     

What is this graph going to look like?

Draw attention to the graph will not deviate up or down from the pattern already established.

6)     

How far will I be from home after the next 1/10 minute? The next 3/10 minute? Does this knowledge change my graph? (No. It just makes explicit something we hadn't noticed yet.)

Knowing an initial condition and knowing the changes is the same thing as knowing all the states.

7)     

How far from home was I one minute prior to 7 minutes? 5 minutes prior? 2/3 minute prior? ½ minute prior? 7/5 minutes prior? (Don't calculate; just write expressions)?

 

8)     

(New phase)

Keep the graph with calculus triangles (don't erase it). Make a table showing a table with the initial information that you have gone 3.8 miles in 7 minutes and you are traveling at a speed of ½ mile/minute. Leave room above and room below for new entries. DON'T CALCULATE ENTRIES. WRITE EXPRESSIONS. Ask the same questions as before. Relate differences between entries in a column to horizontal and vertical sides of calculus triangles in graph.

Represent the above using a table

9)     

Give stu's new numbers (e.g., 17.3 miles after 5 minutes at a speed of 3.2 miles per minute). Ask them to fill in a table (using expressions when they can't calculate) and to make a "pixie dust" graph that includes calculus triangles that reflects the table entries.

Notice that you must have been in an airplane! 3.2 mi/min is 192 mi/hr.

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